For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

learn more… | top users | synonyms (2)

1
vote
1answer
91 views

Hessian matrix as derivative of gradient

From a text: For a real-valued differentiable function $f:\mathbb{R}^n\rightarrow\mathbb{R}$, the Hessian matrix $D^2f(x)$ is the derivative matrix of the vector-valued gradient function $\nabla ...
1
vote
0answers
56 views

Two person zero sum problem, help/guidance needed..

I'm a computer science student and I have this problem I need to solve for my games theory course. I don't have an example to follow, or use as guidance, and my colleagues are not very helpful( as in, ...
0
votes
1answer
17 views

Understanding the basis term

Consider: $$\left( {\matrix{ 0 & 1 & 2 \cr 0 & 0 & 0 \cr } } \right)$$ I want to find a basis for the row-space of the matrix above. One might say $$B = \left\{ {\left( ...
0
votes
2answers
279 views

How do I show this

Given invertible matrices $A,B$ and $P$ such that $A = PB$, then we say that $A$ is left equivalent to $B$. Show that left equivalence is indeed an equivalence relation.
1
vote
2answers
68 views

Why is it true that for every eigen value a: rank $(T^*- a^*I) =$ rank $(T-aI)$?

I've seen a lemma: rank$(T^*- a^*I) =$ rank$(T-aI)$ where $T^*$ is $T$ adjoint and $a^*$ is the adjoint eigen value. I know that $T$ and $T^*$ has the same rank, but can someone help me understand ...
3
votes
0answers
69 views

Continuous subgroup of SO(3)?

I read from a paperarXiv: cond-mat/0602109 by a theoretical physicist, Prof. Frank Bais, close subgroups of $SO(3)$ is given by ${C_n,D_n,T,O,I,SO(2)\rtimes Z_2}$, where $C_n$ is the cyclic group of ...
0
votes
1answer
57 views

Confusion over Matrix rotation

I want to make a function in C++ that accepts an angle 'a', and a vector 'v' as arguments and returns a matrix. 'a' should represent the amount that is rotated around vector 'v', an arbitrary axis, ...
0
votes
1answer
57 views

Eigenvalues and eigenvectors general $n \times n$

Find the eigenvalues and eigenvectors for the general $n \times n$ matrix which has $2$'s across the main diagonal, $-1$'s below and above the main diagonal?
2
votes
1answer
440 views

Notation for the set of symmetric matrices and symmetric positive definite matrices

I would like to know if there exists a notation for the set of symmetric matrices and symmetric positive definite matrices. For instance, the set of $N \times N$ matrices with real entries is denoted ...
0
votes
1answer
33 views

Proof regarding effect of row operations on determinants>

Let $A,B \in K^{n,n}$ and suppose $B$ is obtained from $A$ by adding $\lambda$ times row $j$ to row $i$. Prove $det(A)=det(B)$. My Attempt I tried to use proof by induction for this . Take ...
0
votes
0answers
47 views

Simultaeous diagonalizability and the commutator

Is there any intuitive, down-to-earth, reason why the vanishing of the commutator $[ \ , \ ]$ of two operators $A$ and $B$, $[A,B] = 0$, implies simultaneous diagonalizability of $A$ and $B$, and why ...
1
vote
1answer
43 views

Subgroups of GL(2,C) isomorphic to Z

Let $\mathbb Z\to \mathrm{GL}_2(\mathbb C)$ be an injective homomorphism. I'm wondering about the possibilities for the image of $\mathbb Z$. I think the image is always conjugate to a subgroup of ...
0
votes
1answer
44 views

Solve Matrix: How many trips can be made?

We have the matrix: $$\mathbf{M}=\begin{bmatrix}0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0\end{bmatrix}$$ The matrix ...
1
vote
0answers
112 views

Meaning of $A^A$

Let $A$ be a matrix. I know that there is definition for $A^k$ power of $A$, and $e^A$ expontential of $A$. Is there any meaning of $A^A$?
0
votes
2answers
51 views

Rotation counterclockwise

Let $A_{\theta}$ be rotation counterclockwise by $\theta$ as follows: $$A_\theta = \left[ \begin{matrix} \cos(\theta) & -\sin(\theta)\\ \sin(\theta) & \cos(\theta)\end{matrix} \right]$$ ...
0
votes
2answers
328 views

inequality with the Frobenius norm for matrices

Let $A\in M_n$. How can I show that $$\left|{\textrm{Tr}(A)\over\sqrt{n}}\right|\leq \Vert A\Vert_F$$ I tried it using the Cauchy-Schwarz inequality.
0
votes
0answers
52 views

Number of combinations in a matrix

Given the size of a matrix is $N \times N$, how many unique matrices are there given the following restrictions: Matrix entries can only contain numbers $\left[0,b\right]$ A valid matrix cannot have ...
1
vote
1answer
48 views

linear transformations with matrices $A, A^*$

Let $K$ be a field, $K\subseteq \Bbb C$. $V$ is a linear space over $K$, $\dim(V)=n(n\geq2)$. Choose ordered basis $\epsilon_1,\epsilon_2,\dotsc,\epsilon_n$ for $V$. $\bf A,B$ are two linear ...
1
vote
1answer
82 views

Orthogonal projection of a vector $x$ onto $\operatorname{Col}(A)$

Let $A$ be a matrix. The orthogonal projection of a vector $x$ onto $\operatorname{Col}(A)$ is unique if and only if the columns of $A$ are linearly independent. True or False?
4
votes
2answers
79 views

Compute $ \lim_{n\to \infty}\prod_{i=1}^n B(p_i^{-2})$

Let $B(x) = \begin{pmatrix} 1 & x \\x & 1 \end{pmatrix}$, and $2=p_1<p_2<\cdots <p_n <\cdots$ primes number. Compute $$\displaystyle \lim_{n\to \infty}\prod_{i=1}^n ...
0
votes
0answers
29 views

product of Mueller matrices

I am reading very old Fortran77 program. In the program, there is a subroutine for the product of two 4 x 4 Mueller Matrices. The subroutine produces matrix "C" using matrix "A" and matrix "B". The ...
1
vote
0answers
37 views

spectrum of a special class of tridiagonal matrices

Consider a real and symmetric tridiagonal matrix with zero diagonals and where subdiagonals and superdiagonals are equal to 1 except the (1,2)-th component being equal to $a$, i.e., $\begin{bmatrix}0 ...
0
votes
1answer
16 views

A matrix to form a shorter vector from a given vector by discarding some elements of the latter.

I will try to explain by an example: Given: A 16x1 vector $ V = [a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p,]^T$ and four 9x1 vectors $R_1,R_2,R_3$ and $R_4$. Required: Four 9x16 extraction ...
12
votes
5answers
324 views

Prove that $A^k = 0 $ iff $A^2 = 0$

Let $A$ be a $ 2 \times 2 $ matrix and a positive integer $k \geq 2$. Prove that $A^k = 0 $ iff $A^2 = 0$. I can make it to do this exercise if I have $ \det (A^k) = (\det A)^k $. But this ...
2
votes
2answers
218 views

Find the Norm of Matrix using Cauchy-Schwarz inequality

Let $A$ be $n \times n$ matrix and such that all of its entries are uniformly $O(1)$. Using Cauchy-Schwarz inequality, show that the operator norm of matrix $A$, which is $\|A\|_{op}=\sup_{x\in R^n: ...
2
votes
1answer
65 views

Necessary condition of $|\sum a_n|=\sum |a_n|$

Let $A\in M_{n\times n} (\mathbb{R})$ in which each entry is positive. Let $v$ be a nonzero vector in $\mathbb{C}^n$ where $v_k\neq 0$ Suppose $|\sum_{j=1}^n A_{kj} v_j|=\sum_{j=1}^n |A_{kj} v_j|$. ...
3
votes
0answers
36 views

Problem about real square matrix with rank 1 [duplicate]

Given $A \in \mathbb{R}^{n \times n}$ and $\text{rank}(A) = 1$. By working only on real field, show that $A$ is diagonalizable if and only if $\text{tr}(A) \neq 0$. Here, $\text{tr}(A)$ is the sum of ...
1
vote
1answer
54 views

Matrix over -1/2 [duplicate]

I know what is inverse (-1) of a matrix but I do not have any idea how to compute this P^{-1/2}.
1
vote
1answer
45 views

Prove $A(A+B)^{-1}B = B(A+B)^{-1}A$ where $(A+B)^{-1}$ and $A$ and $B$ are $n\times n$ matricies.

Hi I have been working on this problem for the longest time. Prove: $ A(A+B)^{-1}B = B(A + B)^{-1}A$ We know that A & B exist in real space, and that they are also N x N matrices. It is also ...
0
votes
1answer
28 views

$P$ and $A$ are $3 × 3$ real matrices such that $PAP^t = -A^t$

Let $P$ and $A$ be $3 × 3$ real matrices such that $PAP^t = -A^t$, where $P^t$ denotes the transpose of $P$. Then find det$(P)$. My approach: det$(PAP^t)$ = det$(P)$det$(A)$det$(P^t)$ ...
2
votes
0answers
29 views

Matrix inequality, now what?

I am looking for some motivations. I am reading a book about matrix inequality. e.g. Courant–Fischer equality, eigenvalue of sum of two matrix, ... My question is, so what kind of problems I should ...
0
votes
1answer
88 views

Find the eigenvalues and eigenvectors with zeroes on the diagonal and ones everywhere else.

I have been working on this problem for a couple hours and am completely stuck. Any help would be greatly appreciated. Let $A$ be the $n \times n$ matrix which has zeros on the main diagonal and ones ...
1
vote
1answer
41 views

How to evaluate a limit that involves matrices

I've stumbled upon this problem while I was browsing through the contents of an admission exam . I've struggled tremendously with this exercise and I've got no idea what do to next , it's eating me ...
1
vote
1answer
131 views

Upper bound on the inverse of a Grammian matrix

I have been trying to find a reasonable upper bound on the following: Given $n\in N$ and the Grammian matrix $A_n$ = (($f(i)$ , $f(j)$)) , $f(\lambda) = e^{\lambda t}$ for $0\le t \le 1$ and ...
0
votes
1answer
91 views

Gaussian elimination with partial pivoting with dominant diagonal

Why is it that there is no need for row replacement when performing Gaussian elimination with partial pivoting on a matrics with a dominant diagonal?
2
votes
3answers
2k views

find all values of k for which A is invertible

$\begin{bmatrix} k &k &0 \\ k^2 &2 &k \\ 0& k & k \end{bmatrix}$ what I did is find the det first: $$\det= k(2k-k^2)-k(k^3-0)-0(k^3 -0)=2k^2-k^3-k^4$$ when $det = 0$ ...
1
vote
0answers
77 views

Measure of closeness of a matrix to triangular form

Given a square $n\times n$ matrix $A$, I want to develop a measure of how close the matrix $A$ is to a triangular form.
4
votes
3answers
96 views

Matrix inequality for square matrices

Does the following hold for any square matrix $A$, $(AA^*)^{1/2}\geq (A+A^*)/2$, where superscript $*$ denotes the Hermitian transpose. Proof/any comment would be appreciated.
1
vote
1answer
38 views

Is this matrix-vector equation with given properties always solvable?

Consider a standard n-dimensional matrix-vector equation with a square matrix, $A\textbf{x} = \textbf{b}$ The matrix $A$ is not precisely known in advance, but it is known to have the following ...
6
votes
1answer
2k views

How to prove that the inverse of a matrix is unique?

The ring of matrix is not an integral domain. How to prove that the inverse is unique?
0
votes
1answer
45 views

Prove a statement for the infinite matrix

We are given infinite two dimensional matrix $\{a_{i,j}\}_{i,j=1}^\infty$. And we know that matrix contain only natural values and each number appears in the matrix exactly 8 times. Task is to prove ...
1
vote
1answer
127 views

Proof about weakly elementary matrices;

A square matrix is said to be $weakly \ elementary$ if it is either elementary or it is obtained from an identity matrix by replacing one diagonal entry by zero. Prove that every square matrix is ...
1
vote
0answers
26 views

Algorithm for finding only the $k$-th singular vectors

I know that we have truncated SVD that can compute the first say $k$ largest singular values (and corresponding singular vectors). However, I'd like to know if there is an algorithm that can find only ...
1
vote
0answers
396 views

What is the operation inverse to vectorization (vec operator)?

There is a well knows vectorization operation in matrix analysis $\mbox{vec}$: https://en.wikipedia.org/wiki/Vectorization_%28mathematics%29 I've vectorized my matrix equations, did some ...
0
votes
1answer
605 views

Hermitian matrix has orthogonal eigenvectors for distinct eigenvalues - Proof Strategy [Lay P397 Thm 3]

Herein, I denote the Hermitian conjugate by * (ie: $A* = \bar{A}^T) $. Let $v_i$ and $v_j$ be two eigenvectors of an Hermitian matrix H. First of all suppose that their respective eigenvalues i and j ...
5
votes
3answers
1k views

Reason for reversing the order when transpose and inverse of a group of matrices

Whenever there is a transpose or inverse of a group of matrices, I just reverse their order. For eg: $(ABC)^{-1} = C^{-1}B^{-1}A^{-1}$ and $(ABC)^{T} = C^{T}B^{T}A^{T}$ But usually, I am taking this ...
7
votes
1answer
328 views

Prove that a sequence defined by a recurrence relation converges

Consider the following recurrence relation: $$ a_i = \frac{i+2}{2} \cdot \left(\frac{i}{i+1} - \sum_{j=1}^{i-1} \frac{2 a_j}{2i - j + 2}\right). $$ The first ten terms are: $0.75$ ...
2
votes
0answers
63 views

Best way to solve specific block-tridiagonal linear system (10000x10000 and larger)

To provide more context, this system came from energy balance equation on a mesh with (n,m) nodes in each direction. It's a linear system that looks like this (size of system in blocks n = 4, size of ...
7
votes
2answers
127 views

How to find determinant of this matrix?

Is there a manual method to find $\det\left(XY^{-1}\right)$ ? Let $$X=\left[ {\begin{array}{cc} 1 & 2 & 2^2 & \cdots & 2^{2012} \\ 1 & 3 & 3^2 & \cdots & 3^{2012} \\ ...
1
vote
1answer
52 views

Prove that $ A^2=0 \Leftrightarrow \mbox{ Row}A⊥\mbox{Col}A$

I'd like to get some help So I need to prove that when $A^2=0 \Leftrightarrow \mbox{ Row}A⊥\mbox{Col}A$ Linear Algebra, of course. Thanks